The functorial analytic tropicalization map \(\operatorname{trop}^{an}_{\mathcal {X}}:\vert \mathcal {X}^{\beth } \vert \rightarrow \overline{\Sigma }(\mathcal {X})\)  is defined as follows. Consider a point \([\psi :\operatorname{Spec}R\rightarrow \mathcal {X}]\)  of \(\vert \mathcal {X}^{\beth } \vert \) , where \(R\)  is an integral domain which is complete with respect to a rank-1 valuation \(\operatorname{val}_R:R\rightarrow \mathbb {R}_{\ge 0}\cup \lbrace \infty \rbrace \) . Let \(x\)  be the image via \(\psi \)  of the closed point of \(\operatorname{Spec}R\)  and let \(W_i\)  be the toroidal stratum containing \(x\) . Since \(E_{\mathcal {X}}\)  is étale locally constant along each stratum \(W_i\)  (see [1]}), any étale specialization \(s: w_i\rightsquigarrow x\)  induces by pull-back an isomorphism \(s^*:E_{\mathcal {X},x}\xrightarrow{}E_{\mathcal {X},w_i}\) , which is well-defined up to the action of the monodromy group \(H_{W_i}\)  on \(E_{\mathcal {X},w_i}\) .
Fix an étale specialization \(s: w_i\rightsquigarrow x\)  and consider the chain of monoid homomorphisms
\(E_{\mathcal {X},w_i}\xrightarrow[\cong ]{(s^*)^{-1}}E_{\mathcal {X},x} \xrightarrow{} \big ((R\setminus R^*)/R^*, \cdot \big ) \xrightarrow{} \mathbb {R}_{>0}\cup \lbrace \infty \rbrace \) 
